On Estimate of Settling-Time Distributions of Finite-Time Stable Stochastic Systems

This study characterizes the settling time of stochastic systems with finite-time stability. Finite-time stability guarantees finite-time convergence of system states to an equilibrium. One of the primary features of finite-time stability is settling time, i.e., the time required for the convergence of system states. In stochastic systems, settling time is a random variable and must be characterized stochastically. In this study, the probability distribution of random settling time is estimated. This complements the estimation of settling time in deterministic systems. A lower bound to the probability of settling time is obtained by combining a comparison theorem for stochastic differential equations (SDEs) with a partial differential equation (PDE) that describes the distribution of stopping times. Besides theoretical characterization, a numerical example is also presented to validate the results.